# Properties

 Label 4730.2.a.r Level 4730 Weight 2 Character orbit 4730.a Self dual yes Analytic conductor 37.769 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$4730 = 2 \cdot 5 \cdot 11 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4730.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.7692401561$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta q^{3} + q^{4} - q^{5} + \beta q^{6} + ( -2 - \beta ) q^{7} + q^{8} + ( 2 + \beta ) q^{9} +O(q^{10})$$ $$q + q^{2} + \beta q^{3} + q^{4} - q^{5} + \beta q^{6} + ( -2 - \beta ) q^{7} + q^{8} + ( 2 + \beta ) q^{9} - q^{10} - q^{11} + \beta q^{12} -4 q^{13} + ( -2 - \beta ) q^{14} -\beta q^{15} + q^{16} + ( 1 - \beta ) q^{17} + ( 2 + \beta ) q^{18} + ( 3 - \beta ) q^{19} - q^{20} + ( -5 - 3 \beta ) q^{21} - q^{22} + 4 q^{23} + \beta q^{24} + q^{25} -4 q^{26} + 5 q^{27} + ( -2 - \beta ) q^{28} + ( -2 - 2 \beta ) q^{29} -\beta q^{30} -4 q^{31} + q^{32} -\beta q^{33} + ( 1 - \beta ) q^{34} + ( 2 + \beta ) q^{35} + ( 2 + \beta ) q^{36} -4 q^{37} + ( 3 - \beta ) q^{38} -4 \beta q^{39} - q^{40} + ( 2 - 2 \beta ) q^{41} + ( -5 - 3 \beta ) q^{42} + q^{43} - q^{44} + ( -2 - \beta ) q^{45} + 4 q^{46} + ( -2 + \beta ) q^{47} + \beta q^{48} + ( 2 + 5 \beta ) q^{49} + q^{50} -5 q^{51} -4 q^{52} + ( 1 - 3 \beta ) q^{53} + 5 q^{54} + q^{55} + ( -2 - \beta ) q^{56} + ( -5 + 2 \beta ) q^{57} + ( -2 - 2 \beta ) q^{58} + ( -8 + 3 \beta ) q^{59} -\beta q^{60} + ( 6 - 2 \beta ) q^{61} -4 q^{62} + ( -9 - 5 \beta ) q^{63} + q^{64} + 4 q^{65} -\beta q^{66} -8 q^{67} + ( 1 - \beta ) q^{68} + 4 \beta q^{69} + ( 2 + \beta ) q^{70} + ( -14 + \beta ) q^{71} + ( 2 + \beta ) q^{72} + ( -2 - 2 \beta ) q^{73} -4 q^{74} + \beta q^{75} + ( 3 - \beta ) q^{76} + ( 2 + \beta ) q^{77} -4 \beta q^{78} + ( -6 - \beta ) q^{79} - q^{80} + ( -6 + 2 \beta ) q^{81} + ( 2 - 2 \beta ) q^{82} + ( 4 + \beta ) q^{83} + ( -5 - 3 \beta ) q^{84} + ( -1 + \beta ) q^{85} + q^{86} + ( -10 - 4 \beta ) q^{87} - q^{88} + ( -6 + 6 \beta ) q^{89} + ( -2 - \beta ) q^{90} + ( 8 + 4 \beta ) q^{91} + 4 q^{92} -4 \beta q^{93} + ( -2 + \beta ) q^{94} + ( -3 + \beta ) q^{95} + \beta q^{96} + ( -2 + 4 \beta ) q^{97} + ( 2 + 5 \beta ) q^{98} + ( -2 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} + q^{3} + 2q^{4} - 2q^{5} + q^{6} - 5q^{7} + 2q^{8} + 5q^{9} + O(q^{10})$$ $$2q + 2q^{2} + q^{3} + 2q^{4} - 2q^{5} + q^{6} - 5q^{7} + 2q^{8} + 5q^{9} - 2q^{10} - 2q^{11} + q^{12} - 8q^{13} - 5q^{14} - q^{15} + 2q^{16} + q^{17} + 5q^{18} + 5q^{19} - 2q^{20} - 13q^{21} - 2q^{22} + 8q^{23} + q^{24} + 2q^{25} - 8q^{26} + 10q^{27} - 5q^{28} - 6q^{29} - q^{30} - 8q^{31} + 2q^{32} - q^{33} + q^{34} + 5q^{35} + 5q^{36} - 8q^{37} + 5q^{38} - 4q^{39} - 2q^{40} + 2q^{41} - 13q^{42} + 2q^{43} - 2q^{44} - 5q^{45} + 8q^{46} - 3q^{47} + q^{48} + 9q^{49} + 2q^{50} - 10q^{51} - 8q^{52} - q^{53} + 10q^{54} + 2q^{55} - 5q^{56} - 8q^{57} - 6q^{58} - 13q^{59} - q^{60} + 10q^{61} - 8q^{62} - 23q^{63} + 2q^{64} + 8q^{65} - q^{66} - 16q^{67} + q^{68} + 4q^{69} + 5q^{70} - 27q^{71} + 5q^{72} - 6q^{73} - 8q^{74} + q^{75} + 5q^{76} + 5q^{77} - 4q^{78} - 13q^{79} - 2q^{80} - 10q^{81} + 2q^{82} + 9q^{83} - 13q^{84} - q^{85} + 2q^{86} - 24q^{87} - 2q^{88} - 6q^{89} - 5q^{90} + 20q^{91} + 8q^{92} - 4q^{93} - 3q^{94} - 5q^{95} + q^{96} + 9q^{98} - 5q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.79129 2.79129
1.00000 −1.79129 1.00000 −1.00000 −1.79129 −0.208712 1.00000 0.208712 −1.00000
1.2 1.00000 2.79129 1.00000 −1.00000 2.79129 −4.79129 1.00000 4.79129 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4730.2.a.r 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.r 2 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$1$$
$$43$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4730))$$:

 $$T_{3}^{2} - T_{3} - 5$$ $$T_{7}^{2} + 5 T_{7} + 1$$ $$T_{13} + 4$$